I'm a seasoned electrical engineer, and during my studies I've seen the Maxwell equations in a lot of courses. First in high school, then in the physics course on electromagnetism, then in a dedicated course on antennas and radio wave propagation, then in a course on lasers, then in fundamentals of quantum mechanics and in many other courses. This is the best lecure about he topic that I have ever seen!

My regret is that at 74 years of age I am now finding these wonderful lectures from an enthusiastic, vibrant and delightful teacher . Would I were 20 again. tThank you Kathy.

Kathy takes all the tension away from learning physics and math in a way I have never ever seen before. Showing us why and helping understand how Maxwell's equations came to be, all with an honest love of the subject matter sounds impossible but that is what Kathy does. It is an amazing gift. Thank you.

Fabulous presentation of Maxwell's equations. It brought back memories of deriving them from first principles as an electrical engineering major 40 years ago. I particularly liked how you included images from his papers that do not use vector notation.

I am a recently retired electrical engineer and I would like to say that this is on of the best explanations on the subjet I have ever seen. My congratulations from Spain for your excellent work on your KZfaq channel. By the way, I'm looking forward to reading your book "The Lightning Tamers" which will no doubt be just as fascinating as the videos.

This is by far the best explanation of this that I have ever seen. You managed to get so much into only 34 minutes, I am truly amazed. Especially how you linked all the discoveries.

This thing remained sort of a mystery to me for about three decades. Now I believe I've seen the light. Thank. You.

That was excellent. I wish this had been available when I was studying this stuff 45 years ago. I knew several bits and pieces, and you've assembled them into a consistent story, which triggers more insights. I can re-watch this several times, and pick up a little more each time. Wouldn't it be fun to bring these fellows back in a time machine (especially Hertz, who foresaw no practical uses) and show them what we do with their discoveries today.

Your vision of showing the history of each one of the ideas that finally were mature enough to bring Maxwell to write them with math equations is really brilliant and very powerful to give us a better understanding of this subject. Congratulations for your work. All of these famous developers would be proud of you. From Brazil, a great hug!! Miguez

Wonderfully explained! Studied these equation in first year physics during under graduation and always had these unsettling feeling of not grasping them fully. Over a period of time, have tried to understand from different literatures and physics videos like this in leisure, but so far have found it just mathematical jugglery. This is the first video that I could relate to and ties everything that I know about magnetism and electricity seamlessly. Thank you very much!

At about 14:52, "french polymorph" probably wanted to be polymath, but I'm enjoying the idea of ampere as a shapeshifter.

My son is starting his freshman year in college as a physics major (as his dad did) and I have pointed him at your channel. You are truly a gifted teacher!

I studied physics 40 years ago at King's College London, Maxwell's Alma mater. I remember having such a hard time of it with vector analysis. That green Schaum textbook haunts my memories. I never really understood it - I just memorised it all. No such thing as KZfaq and teachers like Kathy and 3blue1brown. I'm really enjoying this series.

Maxwell, the most underrated physicist of all time. United two separate forces into one mathematically on his own and proved that light was an electromagnetic wave. Even Einstein saw him as a role model.

Your enthusiasm is infectious! While I enjoy math, I am not very gifted or fast at it. Most of the math in my electrical apprenticeship was above my full understanding, but I found that treating it like a language really helped. I have forgotten most of it, but so much enjoy your putting together the historical development of the people and the math that have given us so much in our ability to understand our world.

I must say, you really know how to keep a viewer’s interest in a subject, Miss Kathy. Great job on your latest effort. Thanks so much.

I know you’ve done at least one video on the class and background differences between Faraday and Maxwell, and the general resistance of the aristocratic scientists of the day to properly credit Faraday. So it was nice to see Hertz fully crediting “Faraday-Maxwell” in that paper. Yeah, the science is always cool. But sometimes the personal stories behind the science are equally interesting.

I have bought your book by preorder and a few minutes ago it is delivered to me! Live long and prosper, ma’am!

Thank you for the wonderful content. I’m 57 and still enjoy learning. Your videos are accurate and not dumbed down. You keep me thinking. Thanks again.

Kathy, where were you when I was an undergrad? If these issues had been explain with this clarity, my life would have been MUCH happier. Nice job!

Your videos always make me want to learn more, even when I think I know the subject well.

One of my favorite moments in college physics was when we combined Maxwell's equations and it turns into a wave equation with the constants in the place of velocity, adding more evidence light is composed of electric and magnetic waves.

Well Kathy, I'm just a software engineer that learned Maxwell equations at university, but I must say your video was so enjoyable to me like listening a classical symphony in four parts, with an apotheotic conclusion! Congratulations and all the best!

Your balance of the history behind and not, “dumbing down,” the maths is beautifully unique! Thank you.

I have a Ph.D in EE. My specialty is Telecommunication. I am retired now. Of course, I took grduate courses in EM and antenna theories in one of the good EE Deparment in the US (Polytechnic/NYU). I did not have a textbook that went into the background of these 4 equations, and the professors who lectured in this subject to teach as simplified as Kathy did. It was always bogged down with heavy math without touching the Physical aspects of the phenomena. Students are forced to memorize them even in fine instutions without capturing the essence of them. I do not know your background, Kathy. I wish I had someone like you then to simplify that genius work.

This is really amazing, I usually read history before studying any theory, to learn which events made discoveres to assert something about the theory, But unfortunately, I don't know where to look for proper information and it takes so much time, But here, In one video, I learn your research, which is tedious in real, for the history of the events and also detailed implications ,, This is just amazing ma'am, Thank you for these videos 😊

I've seen many videos on vector multiplication and the divergence and curl operations, but yours finally made it "click." Thanks for being such a good teacher, Kathy.

The best explanation of Maxwell's equation. Loved the part which states "what the equation (statement) means". Also, the historical contexts were quite interesting. Thanks for posting such detailed informative session.

Hi Kathy, one of your patrons, will have this as required watching for my students

I love your videos. I am a biochemist, but I find physics fascinating. If I was more intelligent I would have went into physics. At least I can watch these videos from you and gain a better understanding, you make it so accessible!

what an impressive and important video!!!!! Amazing work and a gift to us Luddites! Am now retired but have worked with so many that didn't know 'the story' about why these are so important today in the way they are. Electromagnetism and algebra Cadabra. Brilliant. The universerse is Gaussian.

I cannot wait to watch the Sir William Rowan Hamilton video! Last month I purchased an original Introduction To Quaternions by Kelland and Tait 1873. Honestly, I have been on one Gnarly ride learning from and reading about the many great minds that once walked the halls at the University of Edinborough. Sooo, Freaking Many! Geez... Last week, I felt alone on this journey. This week, here you go again bringing the heat! Gracias Side note: Oliver Heaviside -NATURE -1893 - Vectors Versus Quaternions I found a hysterical article from Nature by Oliver Heaviside in 1893 with regard to Maxwell's use of quaternions. It's not at all the way I remembered learning about the Heaviside story nor how history has framed him as the first-rate oddity" and Maxwellian Fanboy #1. That was definitely not the case. He's an insult guru of the highest degree..lol I mention it because it relates to your content and is one amusing read. Cheers

Thank you Kathy! For weird reasons, I skipped physics in college. This would likely have been day 1, and I have been mystified about Maxwell's Equations for 40 years. Until now. So I am going to buy your book, and maybe more of physics will open up to me.

This is, again, a _very good video!_ Geometric Algebra, also called Clifford Algebra, makes the Maxwell's equations a lot simpler. Especially because with Geometric Algebra you can represent rotations directly, without needing a vector perpendicular to the distance and force vector. A torque is then a rotation given by what is called an outer product: *_Rot_* = *_F_* ^ *_d_* _in the plane of the vectors_ *_F_* and *_d_* . Both the dot product and the outer product, which replaces the cross product are united in a single product, which is called the Geometric Product, from which the magnitudes and the rotation direction of the outer product, and the inner product can be obtained in the following way: Given two vectors *_u_* and *_v_* in space, you can form the geometric product simply by *_uv_* which is not a commutative product. *_uv_* is not equal to *_vu_* .If the vectors *_u_* and *_v_* are parallel, then the Geometric Product is commutative: *_uv_* = *_vu_* . If the vectors *_u_* and *_v_* are perpendicular, then their Geometric Product is anti-commutative: *_uv_* = - *_vu_* . In all other cases, the Geometric Product is neither commutative nor anti-commutative. The dot product is then given by *_u . v_* = ( *_uv_* + *_vu_* )/2, which is a scalar, and the outer product, which represents a rotation from *_u_* onto *_v_* is then given by *_u ^ v_* = ( *_uv_* - *_vu_* )/2 which is a rotation of *_u_* onto *_v_* . The Geometric Product is then given by *_uv_* = *_u . v_* + *_u ^ v_* In other words, a geometric product is a sum of a scalar and somehting that rotates. A nice thing about the Geometric Product, is that you can both multiply and divide vectors and any formula consisting of sums and geometric products. In other words, Geometric Algebra is a complete algebra, while using scalars, inner products and vector products this cannot be done, because they are all different algebraic entities. Rotations can only be represented by vectors in 3D space. In 1D space you have just a fixed direction, and therefore zero rotation. In the plane you can have just one rotation. And in 4D space you have 6 rotations, not 4! It is just an accident of mathematics, that in 3D space you have as many translations as rotations. This is because any rotation happens in a plane defined by two axes. In 1D space you have only one axis, therefore it is impossible to rotate something in a 1D space. In a 2D space you have two axes, and therefore exactly one kind of rotation. In 3D space there are exactly 3 ways to select 3 axes defining a plane of rotation. That is why you can represent rotations by vectors. But if you do that, you obfuscate, even eliminate _the fundamental difference_ that exists between translations and rotations. In 4D space there are 6 ways to select 2 axes. That is why in 4D space you have 6 rotations. And in 5D space you have 10 rotations, because there are 10 ways to select two axes out of 5 that form a plane within which you can rotate.

Whew - Might need to watch this several times to actually understand such a fundamental mathematical view of physics. Great Video

What a wonderful teacher Kathy is. Wished I had a teacher like her during my undergraduate days.

When I took differential forms my professor Ted Frankel derived some of Maxwell's equations from using the static equations and the Lorentz Transformations as axioms. It was very exciting. I am 75 now and have forgotten some of the details, but it was pretty fun.

Kathy, I think you are explaining the calculus very well of the maxwell equations. Perhaps you can also show examples of what the equations mean in real life eg by showing a magnetic field or induction. Like Maxwell said: what is the go of it. Intuitively understanding what is happening that is more important than the equations themselves.

Fantastic! You are an extremely gifted educator!

Kathy should write a book by putting all the work to gether. I have EE but never made connection on how science evolved and contribution made by others unkowns. As newton once said he was able to see further by standing on shoulder of giants. Maxwell stood on shoulder of faraday, ampere, gauss and columb and may other unkown. Maxwell lived short life but made signeficant contribution and influenced our daily life. Imagine your life without modern communication.

Just to add to the confusion, James Clerk Maxwell also has another set of relations expressed in terms of partial derivatives in thermodynamics that are named after him, i.e., the 'Maxwell Relations'.

It's counter-wise. The changes in the magnetic field changes the distribution of charges, therefore the electric field around them. Great videos, all of them!

After 50 years of mystery Maxwells equations are making sense! And thanks for explaining the T shirt!

Excellent exposition! I would add for those interested, a reference to Oliver Heaviside who according to Wikipedia: " In 1884 he recast Maxwell's mathematical analysis from its original cumbersome form (they had already been recast as quaternions) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell's equations."

The curl of your articulate speech and divergence of knowledge together create dazzling light.. I loved this video.. it is one of those which u will see over and over again.

Amazig lecture. The best I have ever seen. The explanation leaves Feyman in the dust. Thank you.

Her Quaternion lecture is wonderful.

Very conceptual explanation! The historical background and narratives help me understand more the concept. As a graduate of Electrical Engineering, I salute you for tackling this difficult subject. Job well done, Kathy.

Thank you very much for the wonderful lecture, we were never taught like this. Congratulations, Kathy!

Kathy amazing lecture, and the illustrations made it powerful and easier to understand. Thanks

I'm very glad for the scripts. Watching the video's is great, but to really understand these things I have to sit with pencil and paper and write them out. Sometimes I wish I could go back to school.

The subject of this one video would make a great book all by itself. I love how you reference and highlight key sections of original sources in your videos. Great summary of great scientists building on their predecessors work. Thanks so much.

Can't wait for my book!

really accessible, well-explained and has a the right thread to make it easier to follow

You’re one of my favourite professors, Kathy. Thank you for this incredible work.

Beautifully explained and contextualized with historical references to what birthed these equations. Bravo!

kathy you make me love history and physics again..

Techinically del is an operator as it transforms its argument (a multivariable differentiable function) into another object (in this case a vector) according to a well defined law (taking the partial derivatives) The dot product between del and a vector field F is called divergence. Its result is the sum of the partial derivatives of the vector field's components. It represents the tendency of F to clump (negative divergence) orr spread out (positive divergence) in an infititesimal volume dV. If the divergence is 0 then F is said to be solenoidal. The curl on the other hand is the cross product betwwen del and F. It represents the rotantion of F in an infinitesimale volume dV. By convention the positive ditection of Curl(F) corresponds to a couterclockwise rotation of del around F. If the curl is null then F is irrotationak.

Best video I have ever seen about Maxwell’s equations! Respect!

Another electrical engineer. That was an excellent lecture. It has greatly helped my understanding of things I always struggled with. And all done with such infectious enthusiasm. So interesting to hear it through the histroy of how different peoples contributions combined to help advance knowledge.

It is not easy to be an educator. You don’t just tell people stuff. I, like most, suffered through high school and college believing that I just couldn’t understand some things. I have only lately become aware that poor curriculum development and instruction can doom all but the Einsteins and Faradays of the world. We don’t have to be one of these people to understand physics. We just need outstanding teachers like Kathy. I wrote more but deleted it. Being a good teacher is incredibly difficult.

Thanks for making these videos you are making , a little history while learning concepts makes it more interesting. Actually it gives some intuition how those people used to think, what was the reason to do a certain experiment and much more. Thank You !!

Best explanation anywhere! Kathy is both brilliant and sublime. 🎉😊

I have a flue and enjoying your video yet again over a cup if tea.. thank you maam!!

❤❤❤ Thank you! What a wonderful historical overview of my favorite equations ! 👏👏👏👏 I think there is one pearl of inside, regarding this history, that remains usually missing: it was Einstein who, about half a century later, realized that the speed of light must be the same for any inertial reference frame, thus developing special relativity theory. Well, Maxwell didn’t realize it, but In fact special relativity was right in front of his eyes! This is by no means said to diminish the merit of Maxwell, which is immense and calls for all my admiration. The thing is that both permeativity and permisivity are unit dependent constants of proportionality, derived by the measure of the force of a magnetic or a electric field acting on a magnet or electric charge. According to Maxwell’s equations these constants don’t depend by any means on the spatial direction of measurement, nor by the frame of reference in which they are measured. Thus, if the speed of light is the square root of the inverse of the product of two constant numbers, the speed of light must be equally constant, regardless of the reference frame it is measured in!

Oliver Heaviside and Josiah Gibbs were two big proponents of Vector Calculus and writing Maxwell's equations in that notation.

Saving this one to my playlist! Great explanations and refreshers

I like the way you casually refer to Gauss as “a German mathematician.”

Great stuff Kathy. Love your authenticity.

14:53 Ampere was not a “polymorph” (a shape-shifter). He was a “polymath” (a student of diverse subjects). 😊

Kathy, that was a great explanation! And by the way, I love your enthusiasm! Thank you very much

Beautiful presentation!

Lovely explanation, much appreciated. Inwork with ship power generation and distribution plants, and this video makes me feel that the earth itself is a gwnerator, albeit, without an obvious prime mover.

Thanks Kathy for bringing to life all those equations that innervated my engineering schooling and life!

exceptionally thorough presentation and flow. appreciate explaining the difference between mathematical scalar vs vector. so many brilliant minds leading to this understanding. MAXWELL RULES. thank you kindly madame

Recently, I watched animations in several YT videos showing how Fitzgerald contraction of space creates an electric charge density unbalance in a neutral medium with positive and negative charges in equal number if the mean speed of the fore differs from that of the latter, i.e., with an electric current. Such unbalance results in a net divergenceless, curly electric field matching the magnetic field generated by the current. The best explanation I've seen that electric and magnetic phenomena are the same. Apart from the above, your video has answered my years-long question about Maxwell's equations and radiowaves (of any wavelength): whether purely theoretical or based on some measurements, as they turned out to be. Many thanks! Wonderful job! I've listed this video in a very short Physics list for future reference.

I agree with the other commenters that your explanation of Maxwell’s equations coupled with the history of their development is masterful. I’ve learned a lot and my imagination is piqued.

Fantastic presentation! Thank you

What a beautiful summary!! You are an amazing communicator.

Saw and very useful I hope you make this in a playlist

beautiful recap and explanations, thank you. i always recommend your channel to my students.

Thanks for another great video. Waiting for the book to arrive.

Notifications: ON I'm excited about your next video on Quaternions. I was playing with Quaternions while watching your video. What a coincidence. Was trying to make a knot function. I enjoy your videos. They are on point, deep, dense and compressed. You tell things in an inspiring way. Thanks for your hard work. I appreciate it as much as everyone else. A lot.

Kathy, what a great channel, I absolutely love it!!!

I so wish I'd had your videos available when I was taking physics. Good work.

Very Well done ! Hystory citing primary bibliographic sources and mathematical rigor side by side. Fantastic!

This is great! I particularly like the historical approach, showing the original published work of the scientists involved. The names are familar to me, but your video helps us learn the sequence of events, and who influenced whom. Now try doing it again, in spherical coordinates.

I love these videos so much. Thank you for making them.

Very good videos. I check my phone for your postings. And wait in anticipation of there releases. Great job.

Lady ur the best .

Kathy - excellent Job!

Thank you! I am a fanatic of Maxwell's equations and you just gave me the historic perspective and how everything came nicely together. I never had the time to research the history of the equations, and you just served them to me on a silver plate. Keep going the good work.

Dr Kathy is a gifted teacher.

Your videos really click my brain!

this is amazing work, thanks for posting.

Your Videos are amazing Kathy

Thank you so much for this video. I can't quite fiollow the math, but I do follow the conclusions and how they relate. I find it all fascinating!

Love your videos! The history of physics discovers and inventors is amazing! Great job!

Thank you Kathy, that was a great lesson very well explained. You're brilliant!

Maxwell's equations were really the start of modern physics in my opinion. Electricity was so new in the 1860's. Once it began to be understood so much of the rest of what we consider physics today was revealed. But, no one person did it all.